Phase transitions and approximations of mean squared error for state-space models with fractional differencing
Abstract
We study trend estimation in state-space models in which the trend has a fractional stochastic difference of order d>0 and the observation errors form a short-range-dependent stationary process. Using finite-sequence fractional summation and differencing operators, we analyze the penalized least-squares estimator obtained by shrinking the fractional differences of the trend. We derive asymptotic mean squared error (MSE) approximations for all d>0 and identify a sharp phase transition at d=1/2. When d>1/2, the estimator is consistent and its optimally balanced MSE has order n−(2d−1)/(2d). At the boundary d=1/2, we obtain a refined finite-sample approximation and show that the MSE decreases at the slower order loglog n/log n. When 0<d<1/2, the MSE converges to an explicit positive limit, so consistent recovery of the trend is impossible under the considered scaling. We also describe a practical criterion for choosing the penalty parameter and differencing order, and numerical experiments illustrate the MSE approximations and the behavior of the selection procedure.
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BibTeX
@misc{burman2026phase,
title = {{Phase transitions and approximations of mean squared error for state-space models with fractional differencing}},
author = {Prabir Burman and Xiucai Ding and Robert H. Shumway},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06727},
eprint = {2609.06727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06727},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Phase transitions and approximations of mean squared error for state-space models with fractional differencing
AU - Prabir Burman
AU - Xiucai Ding
AU - Robert H. Shumway
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06727
UR - https://arxiv.org/abs/2609.06727
AB - We study trend estimation in state-space models in which the trend has a fractional stochastic difference of order d>0 and the observation errors form a short-range-dependent stationary process. Using finite-sequence fractional summation and differencing operators, we analyze the penalized least-squares estimator obtained by shrinking the fractional differences of the trend. We derive asymptotic mean squared error (MSE) approximations for all d>0 and identify a sharp phase transition at d=1/2. When d>1/2, the estimator is consistent and its optimally balanced MSE has order n^−(2d−1)/(2d). At the boundary d=1/2, we obtain a refined finite-sample approximation and show that the MSE decreases at the slower order loglog n/log n. When 0<d<1/2, the MSE converges to an explicit positive limit, so consistent recovery of the trend is impossible under the considered scaling. We also describe a practical criterion for choosing the penalty parameter and differencing order, and numerical experiments illustrate the MSE approximations and the behavior of the selection procedure.
ER -